MATH1051 Calculus & Linear Algebra 1
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Semester Summer Final Examination, 2020
MATH1051 Calculus & Linear Algebra 1
PART A 20 marks
● Part A is Pass/Fail. A student who Passes Part A earns all 20 marks. A student who Fails Part A earns 0 marks.
● The Pass threshold is 13/20.
● There are 20 questions (each question is worth one mark) on: Gaussian elimination, Inverses, Determinants, Vector Products, Eigenvalues, and Eigenvectors.
1) The augmented matrix of a linear system has been reduced by row operations to the form
How many solutions does this system have?
2) Find the general solution to the following system of equations:
3) Apply elementary row operations until the augmented matrix
is in Gauss reduced form.
4) Determine the nullspace of the following matrix:
5) Apply elementary row operations until the augmented matrix
is in Gauss reduced form.
6) Let A and B be invertible matrices. Is the following statement True?
(AB)−1 = A−1B−1.
7) Let A be a square matrix.
Give three different conditions equivalent to the statement “A is invertible.”
8) Let A be an invertible matrix, and let b be a vector. How many solutions are there to the equation Ax = b?
9) Determine the inverse of the following matrix:
10) Find the determinant of the following matrix:
11) Suppose A and B are matrices, with |A| = 2, and |B| = 5.
Evaluate |AB|.
12) A matrix C has determinant 3.
Which of the following is true? Circle the correct statement:
(a) C is invertible.
(b) C is not invertible.
(c) C may be invertible., but there is not enough information to determine for sure.
13) Given that
14) Let A be a matrix, and suppose |A| = 3.
Evaluate |A−1|.
15) Let u = (0, 1, 1), and v = (1, 0, 3).
Calculate u × v.
16) Suppose u × v = 5j.
Determine (−3u) × v.
17) Let a and b be non-zero vectors, and suppose a × b = 0.
Which of the following is true? Circle the correct statement:
(a) a and b are parallel.
(b) a and b are perpendicular.
(c) a and b may be either parallel or perpendicular, but there is not enough information to determine for sure.
18) Find all eigenvalues for the following matrix:
19) One of the eigenvalues of matrix
Find an eigenvector corresponding to this eigenvalue.
20) One of the eigenvectors of matrix
What is this eigenvector’s corresponding eigenvalue?
PART B - CALCULUS - 40 marks
21) Determine whether the following series converge. Show all working.
22) Recall the double angle formula:
Find the MacLaurin series for f(x) = sin2 x. Show all working.
23) a) Show that
b) Evaluate the volume of the solid obtained by rotating about the x-axis over the interval [0, 1].
24) a) Explain why we would use rule to determine
b) Evaluate (if the limit exists).
PART B - LINEAR ALGEBRA - 30 marks
25) a) Let G be a square matrix such that GGT = I.
Show that |G| = ±1 .
b) A matrix C is called idempotent if C2 = C.
Let A be an m × n matrix and B an n × m matrix.
Prove that if BA = I, then AB is idempotent.
c) Let F be a square matrix. Show that if 3 is an eigenvalue of F, then 9 is an eigenvalue of F2.
26) a) Show that the following set of vectors is not linearly independent:
b) Do the following vectors form a basis for R3?
27) a) It is known that is a vector space.
Find a basis for V. What is dim(V)?
b) Let
Show that W is not a vector space.
2021-11-05